I’ve just seen a theorem about the constant without proof which is:
(DeGiorgi-Nash-Moser Interior Harnack inequality). Let v be a non-negative solution of
$$Lu=div(A(x)\nabla u)=0$$
in $B_1$, then for r < 1
$$sup_{B_r} u ≤ c(1 − r)^{-p} inf_{B_r} u$$
with c,p depending only on n and λ. (For r close to one, we may choose the constant c equal to one by making p large.)
I DO know how to prove it for Laplacian and when it comes to the general elliptic equations I tried to use the weak Harnack’s inequality but I find it’s kind of hard to calculate the constant and didn’t come up with a complete proof. Is there any reference I should read? Moreover, I’m now confused with another question——How much is the constant combined with the shape of the subset V? I do think that there are some results about it, but I don’t know where to learn it.